Studying generalised Thompson's group with tools from geometric group theory and operator algebra

使用几何群论和算子代数的工具研究广义汤普森群

• 批准号: EP/W007371/1
• 负责人: Brita Nucinkis
• 金额: $ 10.14万
• 依托单位: Royal Holloway University of London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW007371%2F1
• 关键词: Studying; generalised; Thompson; group; tools

Many mathematical concepts first arose in descriptions of physical systems, and later took on a life of their own after they proved to be deep and relevant for other areas of mathematics. A remarkable insight of Heisenberg in 1925 suggested that the observables of a quantum system could be realised as infinite matrices satisfying certain commutation relations. This did not really make sense at the the time, but mathematicians quickly developed the necessary tools, and now we know that he was really talking about operators, which are linear transformations on a vector space, and his insight was that this vector space had to be infinite-dimensional. Thus mathematicians were led to the study of operator algebras, which is now a vast area of mathematics that influences many other areas of mathematics, such as group theory, ergodic theory, dynamics, geometric topology, differential topology, noncommutative geometry, logic and set theory, and number theory.We shall study this connection with group theory from a group theorist's point of view: a group is a mathematician's tool to capture the notion of symmetry in the abstract. The study of symmetry provides a powerful guiding principle in a wide varietyof research problems not only in operator algebra, but in many areas of mathematics and the sciences. For that reason applications of groups abound in these fields.The study of examples is essential to the general understanding of the theory. One class of examples in particular are R. Thompson's groups F,T and V and their generalisations, which exhibit some very surprising properties, and, for the past 50 years, have been studied extensively in a wide variety of mathematical subjects: homotopy theory, dynamical systems, infinite simple groups, the word problem, group cohomology, logic and analysis. For instance, Thompson's groups provided the first known examples of infinite, finitely presented simple groups. Since then, Thompson's groups and their various generalisations have generated a large body of research trying to understand their properties, some of which are not completely settled.One powerful approach to generalised Thompson's groups is their description as automorphism groups of certain Cantor algebras; under some mild conditions one can use this viewpoint to apply discrete Morse theory to determine cohomological finiteness properties of these groups. On the other hand, many of the generalised Thompson's groups can be viewed as topological full groups of a Cuntz algebra. This was recently generalised to include groups that are obtained from higher-dimensional graphs. Hence tools including groupoid homology and the K-theory of the groupoid C*-algebra have become available.The purpose of this project is to develop a comprehensive dictionary between the two approaches to be able to answer open questions arising in both fields. For example, we expect to apply Morse theoretic methods to the groups arising from higher rank graphs to determine their cohomological finiteness conditions. On the other hand, tools like groupoid homology promise to be helpful when distinguishing isomorphism types of automorphism groups of Cantor algebras. This project is a feasibility study designed to not only answer questions such as these but also to to develop a far reaching programme for tackling other involved problems from either area.

许多数学概念首先出现在对物理系统的描述中，后来在被证明对数学的其他领域具有深刻性和相关性后，就具有了自己的生命力。海森堡在 1925 年提出的一个非凡见解表明，量子系统的可观测量可以实现为满足某些交换关系的无限矩阵。这在当时并没有真正意义，但数学家很快开发出了必要的工具，现在我们知道他实际上在谈论运算符，即向量空间上的线性变换，他的见解是这个向量空间必须是无限维的。因此，数学家开始研究算子代数，算子代数现在是数学的一个广阔领域，影响着数学的许多其他领域，例如群论、遍历理论、动力学、几何拓扑、微分拓扑、非交换几何、逻辑和集合论我们将从群论学家的角度研究这种与群论的联系：群是数学家捕捉抽象对称概念的工具。对称性的研究不仅为算子代数的各种研究问题提供了强有力的指导原则，而且为数学和科学的许多领域提供了强有力的指导原则。因此，群的应用在这些领域比比皆是。实例研究对于理论的一般理解至关重要。特别是一类例子是 R. Thompson 的群 F、T 和 V 及其推广，它们表现出一些非常令人惊讶的性质，并且在过去 50 年里已经在各种数学学科中得到了广泛的研究：同伦论、动力系统、无限简单群、应用题、群上同调、逻辑与分析。例如，汤普森的群提供了第一个已知的无限、有限简单群的例子。从那时起，汤普森群及其各种推广产生了大量的研究，试图理解它们的性质，其中一些性质尚未完全确定。广义汤普森群的一种有效方法是将其描述为某些康托代数的自同构群；在某些温和的条件下，我们可以利用这一观点应用离散莫尔斯理论来确定这些群的上同调有限性。另一方面，许多广义 Thompson 群可以被视为 Cuntz 代数的拓扑满群。最近这被推广到包括从高维图中获得的组。因此，包括群群同调和群群 C* 代数的 K 理论在内的工具已经可用。该项目的目的是开发两种方法之间的综合字典，以便能够回答两个领域中出现的开放性问题。例如，我们期望将莫尔斯理论方法应用于由更高阶图产生的群，以确定它们的上同调有限性条件。另一方面，像群胚同调这样的工具有望在区分康托代数自同构群的同构类型时有所帮助。该项目是一项可行性研究，旨在不仅回答此类问题，而且还制定一项影响深远的计划，以解决任一领域的其他相关问题。